Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES ASSOCIATING HYPERGEOMETRIC FUNCTIONS
WITH PLANER HARMONIC MAPPINGS
volume 5, issue 4, article 99,
2004.
OM. P. AHUJA AND H. SILVERMAN
Department of Mathematics
Kent State University
Burton, Ohio 44021-9500, USA.
EMail: ahuja@geauga.kent.edu
Department of Mathematics
University of Charleston
Charleston, South Carolina 29424, USA.
EMail: silvermanh@cofc.edu
Received 18 May, 2003;
accepted 30 August, 2004.
Communicated by: N.E. Cho
Abstract
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2000
Victoria University
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Abstract
Though connections between a well established theory of analytic univalent
functions and hypergeometric functions have been investigated by several researchers, yet analogous connections between planer harmonic mappings and
hypergeometric functions have not been explored. The purpose of this paper is
to uncover some of the inequalities associating hypergeometric functions with
planer harmonic mappings.
2000 Mathematics Subject Classification: Primary 30C55, Secondary 31A05,
33C90.
Key words: Planer harmonic mappings, Hypergeometric functions, Convolution multipliers, Harmonic starlike, Harmonic convex, Inequalities.
This research was supported by the University Research Council, Kent State University, Ohio
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1
Positive Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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1.
Introduction
Let H be the class consisting of continuous complex-valued functions which
are harmonic in the unit disk ∆ = {z : |z| < 1} and let A be the subclass of
H consisting of functions which are analytic in ∆. Clunie and Sheil-Small
in [1] developed the basic theory of planer harmonic mappings f ∈ H which
are univalent in ∆ and have the normalization f (0) = 0 = fz (0) − 1. Such
functions, also known as planer mappings, may be written as f = h + g, where
h, g ∈ A. A function f ∈ H is said to be locally univalent and sense-preserving
if the Jacobian J(f ) = |h0 |2 − |g 0 |2 is positive in ∆; or equivalently |g 0 (z)| <
|h0 (z)| (z ∈ ∆). Thus for f = h + g ∈ H we may write
(1.1)
h(z) = z +
∞
X
n=2
n
An z ,
g(z) =
∞
X
Om. P. Ahuja and H. Silverman
Bn z n ,
|B1 | < 1.
n=1
Let SH denote the family of functions h + g which are harmonic, univalent, and
sense-preserving in ∆ where h, g ∈ A and are of the form (1.1). Imposing
the additional normalization condition fz (0) = 0, Clunie and Sheil-Small [1]
0
0
distinguished the class SH
from SH . Both the families SH and SH
are normal
0
families. But, SH is the only compact family with respect to the topology of
locally uniform convergence [1].
∗
Let SH
and KH be the subclasses of SH consisting of functions f which map
∆, respectively, onto starlike and convex domains. If fj = hj + gj , j = 1, 2 are
0
), then we define the convolution f1 ∗ f2 of f1 and f2 in
in the class SH (or SH
the natural way h1 ∗ h2 + g1 ∗ g2 . If φ1 and φ2 are analytic and f = h + g is in
SH , we define
(1.2)
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
fe
∗(φ1 + φ2 ) = h ∗ φ1 + g ∗ φ2 .
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Let a, b, c be complex numbers with c 6= 0, −1, −2, −3, . . . . Then the Gauss
hypergeometric function written as 2 F1 (a, b; c; z) or simply as F (a, b; c; z) is
defined by
F (a, b; c; z) =
(1.3)
∞
X
(a)n (b)n
n=0
(c)n (1)n
zn,
where (λ)n is the Pochhammer symbol defined by
(1.4)
(λ)n =
Γ (λ + n)
= λ(λ + 1) · · · (λ + n − 1)
Γ(λ)
for n = 1, 2, 3, . . . and (λ)0 = 1.
Since the hypergeometric series in (1.3) converges absolutely in ∆, it follows
that F (a, b; c; z) defines a function which is analytic in ∆, provided that c is
neither zero nor a negative integer. As a matter of fact, in terms of Gamma
functions, we are led to the well-known Gauss’s summation theorem: If Re(c −
a − b) > 0, then
(1.5)
F (a, b; c; 1) =
Γ(c)Γ(c − a − b)
, c 6= 0, −1, −2, . . . .
Γ(c − a)Γ(c − b)
In particular, the incomplete beta function, related to the Gauss hypergeometric
function, ϕ(a, c; z), is defined by
(1.6)
ϕ(a, c; z) := zF (a, 1; c; z) =
∞
X
(a)n
n=0
(c)n
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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z
n+1
,
z ∈ ∆, c 6= 0, − 1, − 2, . . . .
Page 4 of 21
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It has an analytic continuation to the z-plane cut along the positive real axis
z
z
from 1 to ∞. Note that ϕ(a, 1; z) = (1−z)
is the
a . Moreover, ϕ(2, 1; z) =
(1−z)2
Koebe function.
The hypergeometric series in (1.3) and (1.6) converge absolutely in ∆ and
thus F (a, b; c; z) and ϕ(a, c; z) are analytic functions in ∆, provided that c is
neither zero nor a negative integer. For further information about hypergeometric functions, one may refer to [2], [6], and [11].
Throughout this paper, let G(z) := φ1 (z) + φ2 (z) be a function where
φ1 (z) ≡ φ1 (a1 , b1 ; c1 ; z) and φ2 (z) ≡ φ2 (a2 , b2 ; c2 ; z) are the hypergeometric functions defined by
(1.7)
φ1 (z) := zF (a1 , b1 ; c1 ; z) = z +
∞
X
(a1 )n−1 (b1 )n−1
n=2
(1.8)
φ2 (z) := zF (a2 , b2 ; c2 ; z) − 1 =
(c1 )n−1 (1)n−1
∞
X
(a2 )n (b2 )n
n=1
(c2 )n (1)n
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
zn,
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n
z ,
a2 b2 < c2 .
It was surprising to discover the use of hypergeometric functions in the proof
of the Bieberbach conjecture by L. de Branges [3] in 1985. This discovery has
prompted renewed interests in these classes of functions. For example, see [7],
[8], and [9].
However, connections between the theory of harmonic univalent functions
and hypergeometric functions have not yet been explored. The purpose of this
paper is to uncover some of the connections. In particular, we will investigate
the convolution multipliers fe
∗(φ1 + φ2 ), where φ1 , φ2 are as defined by (1.7)
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and (1.8) and f is a harmonic starlike univalent (or harmonic convex univalent)
function in ∆.
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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2.
Main Results
We need the following sufficient condition.
Lemma 2.1 ([4, 10]). For f = h + g with h and g of the form (1.1), if
∞
X
(2.1)
n |An | +
n=2
∞
X
n |Bn | ≤ 1,
n=1
∗
then f ∈ SH
.
Theorem 2.2. If aj , bj > 0, cj > aj + bj + 1 for j = 1, 2,, then a sufficient
∗
condition for G = φ1 + φ2 to be harmonic univalent in ∆ and G ∈ SH
, is that
(2.2)
a1 b 1
1+
c 1 − a1 − b 1 − 1
F (a1 , b1 ; c1 ; 1)
a2 b 2
F (a2 , b2 ; c2 ; 1) ≤ 2.
c2 − a2 − b2 − 1
Proof. In order to prove that G is locally univalent and sense-preserving in ∆,
we only need to show that |φ01 (z)| > |φ02 (z)| , z ∈ ∆. In view of (1.7), (1.3),
(1.4) and (1.5) we have
∞
X
(a
)
(b
)
1
n−1
1
n−1
z n−1 |φ01 (z)| = 1 +
n
(c1 )n−1 (1)n−1
>1−
(n − 1)
n=2
Om. P. Ahuja and H. Silverman
+
n=2
∞
X
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
∞
X
(a1 )n−1 (b1 )n−1
(a1 )n−1 (b1 )n−1
−
(c1 )n−1 (1)n−1
(c1 )n−1 (1)n−1
n=2
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=1−
∞
∞
a1 b1 X (a1 + 1)n−1 (b1 + 1)n−1 X (a1 )n (b1 )n
−
c1 n=1 (c1 + 1)n−1 (1)n−1
(c1 )n (1)n
n=1
a1 b1 Γ(c1 + 1)Γ(c1 − a1 − b1 − 1) Γ(c1 )Γ(c1 − a1 − b1 )
·
−
c1
Γ(c1 − a1 )Γ(c1 − b1 )
Γ(c1 − a1 )Γ(c1 − b1 )
a1 b 1
=2−
+ 1 F (a1 , b1 ; c1 ; 1).
c 1 − a1 − b 1 − 1
=2−
Again, using (2.2), (1.5), (1.3), and (1.8) in turn, to the above mentioned inequality, we have
a2 b 2
F (a2 , b2 ; c2 ; 1)
c 2 − a2 − b 2 − 1
a2 b2 Γ(c2 + 1)Γ(c2 − a2 − b2 − 1)
=
c2
Γ(c2 − a2 )Γ(c2 − b2 )
∞
X (a2 )n+1 (b2 )n+1
=
(c2 )n+1 (1)n
n=0
|φ01 (z)| ≥
∞
X
(a2 )n (b2 )n n−1
n
|z|
(c
)
(1)
2
n
n
n=1
∞
X
(a
)
(b
)
2 n 2 n n−1 z = |φ02 (z)| .
≥
n
n=1 (c2 )n (1)n
>
To show that G is univalent in ∆, we assume that z1 , z2 ∈ ∆ so that z1 6= z2 .
Since ∆ is simply connected and convex, we have z(t) = (1 − t)z1 + tz2 ∈ ∆,
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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where 0 ≤ t ≤ 1. Then we can write
Z 1h
i
F (z2 ) − F (z1 ) =
(z2 − z1 ) φ01 (z(t)) + (z2 − z1 ) φ02 (z(t)) dt
0
so that
(2.3)
F (z2 ) − F (z1 )
=
Re
z2 − z1
Z
1
0
Z
z2 − z1
0
0
Re φ1 (z (t)) +
φ2 (z (t)) dt
z2 − z1
1
[Re φ01 (z (t)) − |φ02 (z (t))|]dt
>
0
Om. P. Ahuja and H. Silverman
On the other hand,
Re φ01 (z) − |φ02 (z)|
∞
∞
X
(a1 )n−1 (b1 )n−1 n−1 X (a2 )n (b2 )n n−1
≥1−
n
|z|
−
n
|z|
(c1 )n−1 (1)n−1
(c2 )n (1)n
n=2
n=1
>1−
∞
X
n=2
=2−
(n − 1 + 1)
(a1 )n−1 (b1 )n−1
(c1 )n−1 (1)n−1
∞
X
(a1 )n−1 (b1 )n−1
n=2
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
(c1 )n−1 (1)n−2
−
∞
X
(a2 )n (b2 )n
−
n
(c2 )n (1)n
n=1
∞
X
(a1 )n (b1 )n
n=0
a1 b1
=2− 1+
c 1 − a1 − b 1 − 1
≥ 0, by (2.2).
(c1 )n (1)n
∞
a2 b2 X (a2 + 1)n−1 (b2 + 1)n−1
−
c2 n=1 (c2 + 1)n−1 (1)n−1
F (a1 , b1 ; c1 ; 1) −
a2 b 2
F (a2 , b2 ; c2 ; 1)
c 2 − a2 − b 2 − 1
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Thus (2.3) and the above inequality lead to F (z1 ) 6= F (z2 ) and hence F is
∗
univalent in ∆. In order to prove that G ∈ SH
,using Lemma 2.1, we only need
to prove that
(2.4)
∞
∞
X
(a1 )n−1 (b1 )n−1 X (a2 )n (b2 )n
n
+
n
≤ 1.
(c
)
(1)
(c
)
(1)
1
n−1
n−1
2
n
n
n=2
n=1
Writing n = n − 1 + 1, the left hand side of (2.4) reduces to
∞
a1 b1 X (a1 + 1)n (b1 + 1)n
c1 n=0 (c1 + 1)n (1)n
"∞
#
∞
X (a1 )n (b1 )n
a2 b2 X (a2 + 1)n (b2 + 1)n
+
−1 +
(c1 )n (1)n
c2 n=0 (c2 + 1)n (1)n
n=0
a1 b 1
= F (a1 , b1 ; c1 ; 1)
+1
c1 − a1 − b1 − 1
a2 b 2
+
F (a2 , b2 ; c2 ; 1) − 1.
c 2 − a2 − b 2 − 1
The last expression is bounded above by 1 provided that (2.2) is satisfied. This
completes the proof.
Lemma 2.3 ([5, 10]). For f = h + g with h and g of the form (1.1), if
∞
X
n=2
then f ∈ KH .
2
n |An | +
∞
X
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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2
n |Bn | ≤ 1,
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n=1
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Theorem 2.4. If aj , bj > 0, cj > aj + bj + 2, for j = 1, 2 then a sufficient
condition for G = φ1 + φ2 to be harmonic univalent in ∆ and G ∈ KH , is that
3a1 b1
(a1 )2 (b1 )2
(2.5)
1+
+
F (a1 , b1 ; c1 ; 1)
c1 − a1 − b1 − 1 (c1 − a1 − b1 − 2)2
a2 b 2
(a2 )2 (b2 )2
+
+
F (a2 , b2 ; c2 ; 1) ≤ 2.
c2 − a2 − b2 − 1 (c2 − a2 − b2 − 2)2
Proof. The proof of the first part is similar to that of Theorem 2.2 and so it is
omitted. In view of Lemma 2.3, we only need to show that
∞
X
∞
n2
n=2
(a1 )n−1 (b1 )n−1 X 2 (a2 )n (b2 )n
+
n
≤ 1.
(c1 )n−1 (1)n−1
(c
)
(1)
2
n
n
n=1
That is,
(n + 2)2
n=0
∞
X
(a1 )n+1 (b1 )n+1
(a2 )n+1 (b2 )n+1
+
(n + 1)2
≤ 1.
(c1 )n+1 (1)n+1
(c2 )n+1 (1)n+1
n=0
But,
∞
X
(n + 2)2
n=0
∞
X
Om. P. Ahuja and H. Silverman
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∞
X
(2.6)
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
(a1 )n+1 (b1 )n+1
(c1 )n+1 (1)n+1
∞
∞
X
(a1 )n+1 (b1 )n+1
(a1 )n+1 (b1 )n+1 X (a1 )n+1 (b1 )n+1
=
(n + 1)
+2
+
(c1 )n+1 (1)n
(c1 )n+1 (1)n
(c1 )n+1 (1)n+1
n=0
n=0
n=0
(a1 )2 (b1 )2
3a1 b1
=
+
+ 1 F (a1 , b1 ; c1 ; 1) − 1,
(c1 − a1 − b1 − 2)2 c1 − a1 − b1 − 1
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and
∞
X
(n + 1)2
n=0
=
(a2 )n+1 (b2 )n+1
(c2 )n+1 (1)n+1
∞
X
(a2 )n+1 (b2 )n+1
n=1
(c2 )n+1 (1)n−1
+
∞
X
(a2 )n+1 (b2 )n+1
(c2 )n+1 (1)n
n=0
(a2 )2 (b2 )2
a2 b 2
=
+
F (a2 , b2 ; c2 ; 1) − 1.
(c2 − a2 − b1 − 2)2 c1 − a1 − b1 − 1
Thus, (2.6) is equivalent to
(a1 )2 (b1 )2
3a1 b1
+
+1 −1
F (a1 , b1 ; c1 ; 1)
(c1 − a1 − b1 − 2)2 c1 − a1 − b1 − 1
(a2 )2 (b2 )2
a2 b 2
+ F (a2 , b2 ; c2 ; 1)
+
≤1
(c2 − a2 − b2 − 2)2 c2 − a2 − b2 − 1
which is true because of the hypothesis.
∗
∗
Denote by SRH
and KRH , respectively, the subclasses of SH
and KH consisting of functions f = h + g so that h and g are of the form
(2.7) h(z) = z −
∞
X
An z n ,
g(z) =
n=2
∞
X
Bn z n ,
An ≥ 0, Bn ≥ 0, B1 < 1.
n=1
Lemma 2.5 ([4, 10]). Let f = h + g be given by (2.7). Then
∗
(i) f ∈ SRH
⇔
∞
P
n=2
nAn +
∞
P
n=1
nBn ≤ 1,
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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(ii) f ∈ KRH ⇔
∞
P
n 2 An +
n=2
∞
P
n2 Bn ≤ 1.
n=1
Theorem 2.6. Let aj , bj > 0, cj > aj + bj + 1, for j = 1, 2 and a2 b2 < c2 . If
φ1 (z)
(2.8)
G1 (z) = z 2 −
+ φ2 (z)
z
then
∗
(i) G1 ∈ SRH
⇔(2.2) holds
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
(ii) G1 ∈ KRH ⇔(2.5) holds.
Proof. (i) We observe that
G1 (z) = z −
∞
X
n=2
Om. P. Ahuja and H. Silverman
∞
X
(a1 )n−1 (b1 )n−1 n
(a2 )n (b2 )n n
z +
z ,
(c1 )n−1 (1)n−1
(c2 )n (1)n
n=1
∗
∗
and SRH
⊂ SH
. In view of Theorem 2.2, we only need to show the necessary
∗
∗
condition for G1 to be in SRH
. If G1 ∈ SRH
, then G1 satisfies the inequality
in Lemma 2.5(i) and the result in (i) follows from Lemma 2.5(i). The proof of
(ii) is similar because KRH ⊂ KH , and by using Lemma 2.5(ii) and Theorem
2.4.
Theorem 2.7. Let aj , bj > 0, cj > aj + bj + 1, for j = 1, 2 and a2 b2 < c2 . A
∗
∗
necessary and sufficient condition such that fe
∗(φ1 + φ2 ) ∈ SRH
for f ∈ SRH
is that
(2.9)
F (a1 , b1 ; c1 ; 1) + F (a2 , b2 ; c2 ; 1) ≤ 3,
where φ1 , φ2 are as defined, respectively, by (1.7) and (1.8).
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∗
Proof. Let f = h + g ∈ SRH
, where h and g are given by (2.7). Then
f ∗˜(φ1 + φ2 ) (z) = h(z) ∗ φ1 (z) + g(z) ∗ φ2 (z)
=z−
∞
X
(a1 )n−1 (b1 )n−1
n=2
(c1 )n−1 (1)n−1
n
An z +
∞
X
(a2 )n (b2 )n
n=1
(c2 )n (1)n
Bn z n .
∗
In view of Lemma 2.5(i), we need to prove that f ∗˜(φ1 + φ2 ) ∈ SRH
if and only
if
∞
∞
X
X
(a1 )n−1 (b1 )n−1
(a2 )n (b2 )n
(2.10)
n
An +
n
Bn ≤ 1.
(c1 )n−1 (1)n−1
(c2 )n (1)n
n=2
n=1
Om. P. Ahuja and H. Silverman
As an application of Lemma 2.5(i), we have
1
1
, |Bn | ≤ .
n
n
Therefore, the left side of (2.10) is bounded above by
|An | ≤
∞
X
(a1 )n−1 (b1 )n−1
n=2
(c1 )n−1 (1)n−1
+
∞
X
(a2 )n (b2 )n
n=1
(c2 )n (1)n
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
= F (a1 , b1 ; c1 ; 1) + F (a2 , b2 ; c2 ; 1) − 2.
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The last expression is bounded above by 1 if and only if (2.9) is satisfied. This
proves (2.10) and results follow.
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Theorem 2.8. If aj , bj > 0 and cj > aj + bj for j = 1, 2, then a sufficient
condition for a function
Z z
Z z
G2 (z) =
F (a1 , b1 ; c1 ; t)dt +
[F (a2 , b2 ; c2 ; t) − 1]dt
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∗
to be in SH
is that
F (a1 , b1 ; c1 ; 1) + F (a2 , b2 ; c2 ; 1) ≤ 3.
Proof. In view of Lemma 2.1, the function
G2 (z) = z +
∞
X
(a1 )n−1 (b1 )n−1
n=2
is in
∗
SH
(c1 )n−1 (1)n
zn +
∞
X
(a2 )n−1 (b2 )n−1
n=2
(c2 )n−1 (1)n
zn
if
∞
∞
X
(a1 )n−1 (b1 )n−1 X (a2 )n−1 (b2 )n−1
n
+
n
≤ 1.
(c
)
(1)
(c
)
(1)
1
n−1
n
2
n−1
n
n=2
n=2
That is, if
Equivalently, G ∈
∞
X
(a1 )n (b1 )n
n=1
∗
SH if
(c1 )n (1)n
+
∞
X
(a2 )n (b2 )n
n=1
(c2 )n (1)n
≤ 1.
F (a1 , b1 ; c1 ; 1) + F (a2 , b2 ; c2 ; 1) ≤ 3.
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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Theorem 2.9. If a1 , b1 > −1, c1 > 0, a1 b1 < 0, a2 > 0, b2 > 0, and cj >
aj + bj + 1, j = 1, 2, then
Z z
Z z
[F (a2 , b2 ; c2 ; t) − 1]dt
G2 (z) =
F (a1 , b1 ; c1 ; t)dt +
0
is in
∗
SH
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0
if and only if F (a1 , b1 ; c1 ; 1) − F (a2 , b2 ; c2 ; 1) + 1 ≥ 0.
J. Ineq. Pure and Appl. Math. 5(4) Art. 99, 2004
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Proof. Applying Lemma 2.5(i) to
G2 (z) = z −
∞
∞
|a1 b1 | X (a1 + 1)n−2 (b1 + 1)n−2 n X (a2 )n−1 (b2 )n−1 n
z +
z ,
c1 n=2
(c1 + 1)n−2 (1)n
(c
)
(1)
2
n−1
n
n=2
it suffices to show that
∞
∞
|a1 b1 | X (a1 + 1)n−2 (b1 + 1)n−2 X (a2 )n−1 (b2 )n−1
n
+
n
≤ 1.
c1 n=2
(c1 + 1)n−2 (1)n
(c2 )n−1 (1)n
n=2
Or equivalently
∞
∞
X
(a1 + 1)n (b1 + 1)n
c1 X (a2 )n (b2 )n
c1
+
≤
.
(c
+
1)
(1)
|a
b
|
(c
)
(1)
|a
b
|
1
1
1
2
1
1
n
n+1
n
n
n=0
n=1
But, this is equivalent to
∞
∞
c1 X (a1 )n (b1 )n
c1 X (a2 )n (b2 )n
c1
+
≤
.
a1 b1 n=1 (c1 )n (1)n
|a1 b1 | n=1 (c2 )n (1)n
|a1 b1 |
Inequalities Associating
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That is,
F (a1 , b1 ; c1 ; 1) − F (a2 , b2 ; c2 ; 1) ≥ −1.
This completes the proof of the theorem.
Remark 2.1. Comparable results to Theorems 2.7, 2.8, 2.9 for harmonic convex
functions may also be obtained. The proofs and results are similar and hence
are omitted.
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In particular, the results parallel to Theorems 2.2, 2.4, 2.6 to 2.9 may also be
obtained for the incomplete beta function ϕ(a, c; z) as defined by (1.6). If
ψ1 (z) := zϕ(a1 , c1 ; z) = z +
ψ2 (z) := zϕ(a2 , c2 ; z) − 1 =
∞
X
(a1 )n−1
n=2
∞
X
n=1
(c1 )n−1
zn,
(a2 )n n
z , a2 < c2 ,
(c2 )n
then
ψ1 (z) + ψ2 (z) ≡ φ1 (z) + φ2 (z),
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
whenever b1 = 1, b2 = 1.
Note that
c1
and
(c1 − a1 )
a2
.
ψ2 (1) = F (a2 , 1; c2 ; 1) − 1 =
(c2 − a2 )
ψ1 (1) = F (a1 , 1; c1 ; 1) =
As an illustration, we close this section with the incomplete beta function
analog to some of the earlier results.
Theorem 2.20 . If aj > 0 and cj > aj +2 for j = 1, 2 , then a sufficient condition
∗
for ψ1 + ψ2 to be harmonic univalent in ∆ with ψ1 + ψ2 ∈ SH
is
a22
c1 (c1 − 2)
+
≤ 2.
(c1 − a1 ) (c1 − a1 − 2) (c2 − a2 ) (c2 − a2 − 2)
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Theorem 2.40 . If aj > 0 and cj > aj +3 for j = 1, 2 , then a sufficient condition
for ψ1 + ψ2 to be harmonic univalent in ∆ with ψ1 + ψ2 ∈ KH is
c1
2a2
3a1
1+
+
(c1 − a1 )
c1 − a1 − 2 (c1 − a1 − 3)2
a2
a2
2(a2 )2
+
+
≤ 2.
(c2 − a2 ) c2 − a2 − 2 (c2 − a2 − 3)2
Theorem 2.70 . A necessary and sufficient condition such that f ∗˜(ψ1 + ψ2 ) ∈
∗
∗
SRH
for f ∈ SRH
is that
c1
a2
+
≤ 1.
(c1 − a1 ) (c2 − a2 )
Theorem 2.90 . If a1 > −1, c1 > 0, a1 < 0, a2 > 0, cj > aj + 1 for j = 1, 2,
and cj > aj + bj + 1, j = 1, 2, then
Z z
Z z
[ϕ (a2 , c2 ; t) − 1]dt
ϕ (a1 , c1 ; t)dt +
0
0
∗
is in SH
if and only if
c1 − 1
a2
≥
.
c 1 − a1 − 1
c 2 − a2 − 1
Inequalities Associating
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Om. P. Ahuja and H. Silverman
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2.1.
Positive Order
We say that f of the form (1.1)
is harmonic starlike of order α, 0 ≤ α ≤ 1,for
∂
∗
∗
|z| = r if ∂θ
arg f (reiθ ) ≥ α, |z| = r. Denote by SH
(α)and SRH
(α)
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∗
∗
the subclasses of SH
and SRH
, respectively, that are starlike of order α. Also,
denote by KH (α) and KRH (α) the subclasses of KH and KRH , respectively, that
are convex of order α. Most of our results can also be rewritten for functions of
positive order by using similar techniques. For instance, using the results in [4]
we have the following:
Theorem 2.10. If aj , bj > 0 and cj > aj + 1, a2 b2 < c2 for j = 1, 2, then
∗
φ1 + φ2 is harmonic univalent in ∆ with φ1 + φ2 ∈ SH
(α), 0 ≤ α ≤ 1 if
a1 b 1
F (a1 , b1 ; c1 ; 1)
1−α+
c 1 − a1 − b 1 − 1
a2 b2
+ α+
F (a2 , b2 ; c2 ; 1) ≤ 2(1 − α).
c 2 − a2 − b 2 − 1
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References
[1] J. CLUNIE AND T. SHEIL-SMALL, Harmonic univalent functions, Ann.
Acad. Aci., Fenn. Ser. A. I. Math., 9 (1984), 3–25.
[2] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[3] L. DE BRANGES, A proof of the Bieberbach conjecture, Acta Math., 154
(1985), 137–152.
[4] J.M. JAHANGIRI, Harmonic functions starlike in the unit disk, J. Math.
Anal. Appl., 235 (1999), 470–477.
[5] J.M. JAHANGIRI, Coefficient bounds and univalence criteria for harmonic functions with negative coefficients, Ann. Univ. Mariae Curie –
Sklodowska Sect A., A 52 (1998), 57–66.
Inequalities Associating
Hypergeometric Functions With
Planer Harmonic Mappings
Om. P. Ahuja and H. Silverman
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[6] E. MERKES AND B.T. SCOTT, Starlike hypergeometric functions, Proc.
American Math Soc., 12 (1961), 885–888.
[7] S.S. MILLER AND P.T. MOCANU, Univalence of Gaussian and confluent
hypergeometric functions, Proc. Amer. Math. Soc., 110(2) (1990), 333–
342.
[8] St. RUSCHEWEYH AND V. SINGH, On the order of starlikeness of hypergeometric functions, J. Math. Anal. Appl., 113 (1986), 1–11
[9] H. SILVERMAN, Starlike and convexity properties for hypergeometric
functions, J. Math. Anal. Appl., 172 (1993), 574–581.
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[10] H. SILVERMAN AND E.M. SILVIA, Subclasses of harmonic univalent
functions, New Zealand J. Math., 28 (1999), 275–284.
[11] H.M. SRIVASTAVA AND H.L. MANOCHA, A Treatise on Generating
Functions, Ellis Horwood Limited and John Wiley & Sons, New York,
Chichester, Brisbane, Toronto, 1984.
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